A338809 -id:A338809 - cp's OEIS frontend

A338825 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, created when an n-bipyramid, formed from two n-gonal pyraminds joined at the base, is internally cut by all the planes defined by any three of its vertices.

12, 8, 120, 84, 24, 448, 280, 28, 368, 256, 48, 32, 1332, 1440, 540, 72, 1160, 1380, 500, 220, 40, 40, 2992, 5280, 2816, 748, 44, 3288, 4272, 1608, 672, 192, 7176, 14040, 8684, 3120, 624, 156, 8120, 12460, 7084, 2968, 1064, 532, 84, 14820, 34020, 22620, 7560, 2580, 720, 120

Offset: 3

Scott R. Shannon, Nov 11 2020

See A338809 for further details and images for this sequence.

The author thanks Zach J. Shannon for assistance in producing the images for this sequence.

			The 4-bipyramid (an octahedron) is cut with 3 internal planes defined by all 3-vertex combinations of its 6 vertices. This leads to the creation of 8 4-faced polyhedra. See A338622.
The 7-bipyramid is cut with 36 internal planes defined by all 3-vertex combinations of its 9 vertices. This leads to the creation of 448 4-faced polyhedra, 280 5-faced polyhedra, and 28 6-faced polyhedra, 756 polyhedra in all.
The table begins:
     12;
      8;
    120;
     84,     24;
    448,    280,     28;
    368,    256,     48,    32;
   1332,   1440,    540,    72;
   1160,   1380,    500,   220,    40,   40;
   2992,   5280,   2816,   748,    44;
   3288,   4272,   1608,   672,   192;
   7176,  14040,   8684,  3120,   624,  156;
   8120,  12460,   7084,  2968,  1064,  532,   84;
  14820,  34020,  22620,  7560,  2580,  720,  120;
  18528,  28480,  18560,  9024,  2592, 1024,  384,  64;
  32028,  66708,  51136, 22372,  7956, 1836,  136;
  35280,  53028,  37080, 14364,  4104,  360,  180, 144;
  57380, 131480, 104576, 50616, 17328, 4256,   76;
  69160, 123040,  86240, 46080, 17600, 5920, 1920, 320, 320;

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, 5-bipyramid showing the 120 polyhedra post-cutting and exploded. Each piece has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. All 120 polyhedra have 4 faces, shown in red.
Scott R. Shannon, 5-bipyramid, seen from above, showing the 120 polyhedra post-cutting and exploded.
Scott R. Shannon, 20-bipyramid, showing the 69160 4-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 123040 5-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 86240 6-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 46080 7-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 17600 8-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 5920 9-faced polyhedra.
Scott R. Shannon, 20-bipyramid, showing the 320 11-faced polyhedra
Scott R. Shannon, 20-bipyramid, showing the 1920 10-faced and the 320 12-faced polyhedra. These are colored white and black respectively. These are not visible on the surface of the 20-bipyramid.
Scott R. Shannon, 20-bipyramid, showing all 350600 polyhedra.
Scott R. Shannon, 20-bipyramid from above, slightly exploded, showing the 69160 4-faced polyhedra.
Scott R. Shannon, 20-bipyramid from the side, slightly exploded and colored white, showing the 69160 4-faced polyhedra.
Eric Weisstein's World of Mathematics, Dipyramid.
Wikipedia, Bipyramid.

Cf. A338809 (number of polyhedra), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sum of row n = A338809(n).

A347753 Number of polyhedra formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

96, 2968, 42384, 319416

Offset: 1

Scott R. Shannon, Sep 12 2021

For a row of n adjacent cubes create all possible planes defined by connecting any three of their vertices. For example, in the case of a single cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the entire solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for n adjacent cubes.

See A347918 for the number of k-faced polyhedra for each value of n.

			a(1) = 96. A single cube, with eight vertices, has 14 internal cutting planes resulting in 96 polyhedra. See A333539 and A338571.
a(2) = 2968. Two adjacent cubes, with twelve vertices, have 51 internal cutting planes resulting in 2968 polyhedra.
a(3) = 42384. Three adjacent cubes, with sixteen vertices, have 124 internal cutting planes resulting in 42384 polyhedra.
a(4) = 319416. Four adjacent cubes, with twenty vertices, have 245 internal cutting planes resulting in 319416 polyhedra.

Scott R. Shannon, The 245 cutting planes on the surface of 4 adjacent cubes.
Scott R. Shannon, The surface of the 4 adjacent cubes after cutting. The 4-, 5-, 6-, 7-, 8-, and 9-faced polyhedra created by the planes are colored red, orange, yellow, green, blue, and indigo, respectively. The 10-, 11-, and 12-faced polyhedra are not visible on the surface. See also A347918.
Scott R. Shannon, The 4 adjacent cubes after cutting exploded. Each of the 319416 polyhedra is moved away from the center of the solid a distance proportional to the average distance of its vertices from the center.

Cf. A347918 (number of k-faced polyhedra), A333539 (n-dimensional cube), A338571 (Platonic solids), A338783 (n-prism), A338809 (n-bipyramid), A007588.

a(1) = A333539(3).

Conjectured formula for the number of internal cutting planes for n adjacent cubes is A007588(n+1).

cp's OEIS Frontend

A338825 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, created when an n-bipyramid, formed from two n-gonal pyraminds joined at the base, is internally cut by all the planes defined by any three of its vertices.

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